Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,16,18,20,27,26)(15,19,24,22,25,28,23,21);(5,7,9,8)(6,10,12,11)(13,14,17,16,18,20,27,26)(15,21,23,28,25,22,24,19);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,15)(14,19)(16,22)(17,24)(18,25)(20,28)(21,26)(23,27);(5,8,9,7)(6,11,12,10)(13,16,17,20,18,26,27,14)(15,22,24,28,25,21,23,19)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317243, #72317284, #72317290 |
| maximal normal subgroup of | #72317243, #72317284, #72317290 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286307);